I was checking my directory and found out that the Folder Backup in Library > Application Support > Mobile Sync > Backup, is taking 13GB of my Harddisk ... What is this and why is taking so much space .. Can I delete it ?
It's the folder where iTunes saves the backUps of your iDevice. And it takes so much space because each time you sync a device, previous backUps are not deleted.
Deleting the folder will, then, mean losing those backUps. Doing this regularly is advised for saving HD space. Just make sure to make a new backUp afterwards, and that you won't need those backUps.
You can do that through iTunes too, by:
Note that all of the backUps of that iDevice will be deleted.
I assume that you are referring to ~/Library/Application Support/Mobile Sync/Backup. This is the location where backups of iOS devices created by iTunes reside. Each sub-folder represents a different iOS device. If you delete this folder you will be unable to restore your iOS devices to their backed up state from iTunes. if you wish to see a list of backups stored by iTunes, open iTunes, go to the iTunes menu > preferences. Find the Devices tab in preferences and select it and you should see a list of device backups.